## Linear Algebra with The Eigen C++ Library ## +24 ## DANIEL HANSON ## Outline • A short history – linear algebra and C++ (1998 – Present) • The Eigen C++ Template Library for Linear Algebra • Linear Basics • Using with Eigen ## Disclaimers/Caveats • This presentation is on solving problems using • The Eigen linear algebra library • stdBLAS in C++26 • Not affiliated with Eigen but have used it in History: Open-Source Libraries • “Next Generation” linear algebra libraries (not exhaustive): • Eigen (2006) • Armadillo (2009) • Blaze (2012) • Expression template-based libraries • Include the usual

Concurrency) • Obtain statistical properties of the tumors (Parallel STL algorithms) ## • Eigen (Array Class) • Eigen is a C++ template library for linear algebra: matrices, vectors, numerical solvers, and [Image](/uploads/documents/6/8/9/c/689cb3d2936cf75eb6743854e14aa392/p8_4.jpg) #include <eigen dense=""> class TumorGenerator { Eigen::ArrayXd initial_population; double tmax; double dt; //... Cell dynamics: $$ rowsatcompiletime,="" int="" colsatcompiletime=""> typedef Eigen::Arrayeigen::dynamic,="" 1=""> Eigen::ArrayXd;eigen> ## Transition of types forms a network ![Image](/uplo

ibrary code_generation) add_dependencies(fast_forward_kinematics_library LBFGSpp) find_package(Eigen3 3.3 NO MODULE) target_compile_definitions(fast_forward_kinematics_library PUBLIC "${FAST FK _library code_generation) dd_dependencies(fast_forward_kinematics_library LBFGSpp) ind_package(Eigen3 3.3 NO MODULE) arget_compile_definitions(fast_forward_kinematics_library PUBLIC "${FAST FK NUMBER forward_kinematics_lib.cpp) dd_dependencies(fast_forward_kinematics_library LBFGSpp) ind_package(Eigen3 3.3 NO MODULE) target_compile_definitions(fast_forward_kinematics_library PUBLIC "${FAST FK

nt*, const double*, unsigned int); } ## cpp namespace robot_joint { Eigen::Isometry3d Joint::calculate_transform( const Eigen::VectorXd& variables ) { const auto rust_isometry = robot_joi variables.size()); Eigen::Isometry3d transform; transform.matrix() = Eigen::Map<eigen::matrix4d>(rust_isometry.data); return transform; } // namespace robot_jointeigen::matrix4d>  project(bridge VERSION 0.1.0) find_package(Eigen3 REQUIRED)  ###

(with FILE SET) add_library(Eigen INTERFACE) target_sources(Eigen INTERFACE FILE_SET HEADERS BASE_DIRS src FILES src/eigen.h src/vector.h src/matrix.h install(TARGETS Eigen EXPORT eigenExport FILE_SET HEADERS DESTINATION include/Eigen) install(EXPORT eigenExport NAMESPACE Upstream::DESTINATION lib/cmake/Eigen) add_executable(exe1 exe1.cpp) target_link_libraries(exe1 Eigen) ## Running Install # running t/include/Eigen/eigen.h -- Installing: C:/Users/hoffman/Work/cxxmodules/file_set/b/inst/include/Eigen/vector.h -- Installing: C:/Users/hoffman/Work/cxxmodules/file_set/b/inst/include/Eigen/matrix.h

inline constexpr Eigen::Matrix runge_kutta4(std::function<eigen::matrix(double, Eigen::Matrix) fun, double time, Eigen::MatrixEigen::Matrix stateOut = common::math::runge_kutta4eigen::matrix>(derivFun, currTime, stateIn stateIn, dt);eigen::matrix ## Usage Example Continued template inline constexpr OutputType ru

$$ T_{0}^{N}=\prod_{0}^{N-1}T_{i}^{i+1} $$ ## One way of solving this problem using Transform = Eigen::Isometry3d; using Joint = std::variant; class Fixed { Transform problem class Revolute { Transform fixed_; Eigen::Vector3d axis_; ... Transform pose(double position) const { return fixed_ * Eigen::AngleAxisd(position, axis_; } }; ## One way this problem class Prismatic { Transform pose(double position) const { return fixed_ * Eigen::Translation3d(position * axis_); } }; ### Requisite godbolt.org demo • https://godbolt.org/z/5jM9sa3vY

UniformScaling(2); julia> a = [1 2; 3 4] 2×2 Array{Int64,2}: 83.1. SPECIAL MATRICES 965 Matrix type LAPACK eigen eigvals eigvecs svd svdvals Symmetric SY ARI Hermitian HE ARI UpperTriangular TR A A A UnitUpperTriangular QRPivoted Pivoted QR factoriza�on LQ QR factoriza�on of transpose(A) Hessenberg Hessenberg decomposi�on Eigen Spectral decomposi�on GeneralizedEigen Generalized spectral decomposi�on SVD Singular value decomposi�on a copy. LinearAlgebra.Eigen – Type. Eigen <: Factorization Matrix factoriza�on type of the eigenvalue/spectral decomposi�on of a square matrix A. This is the return type of eigen, the corresponding matrix

julia> U = UniformScaling(2); julia> a = [1 2; 3 4]; 2×2 Array{Int64,2}: |Matrix type|LAPACK|eigen|eigvals|eigvecs|svd|svdvals| |---|---|---|---|---|---|---| |Symmetric|SY||ARI|||| |Hermitian|HE||ARI|||| |QRPivoted|Pivoted QR factorization| |LQ|QR factorization of transpose(A)| |Hessenberg|Hessenberg decomposition| |Eigen|Spectral decomposition| |GeneralizedEigen|Generalized spectral decomposition| |SVD|Singular value copy. LinearAlgebra.Eigen - Type. Eigen <: Factorization Matrix factorization type of the eigenvalue/spectral decomposition of a square matrix A. This is the return type of eigen, the corresponding

julia> U = UniformScaling(2); julia> a = [1 2; 3 4]; 2×2 Array{Int64,2}: |Matrix type|LAPACK|eigen|eigvals|eigvecs|svd|svdvals| |---|---|---|---|---|---|---| |Symmetric|SY||ARI|||| |Hermitian|HE||ARI|||| |QRPivoted|Pivoted QR factorization| |LQ|QR factorization of transpose(A)| |Hessenberg|Hessenberg decomposition| |Eigen|Spectral decomposition| |GeneralizedEigen|Generalized spectral decomposition| |SVD|Singular value copy. LinearAlgebra.Eigen - Type. Eigen <: Factorization Matrix factorization type of the eigenvalue/spectral decomposition of a square matrix A. This is the return type of eigen, the corresponding